On the Exhaustivity of Simplicial Partitioning

Abstract During the last 40 years, simplicial partitioning has shown itself to be highly useful, including in the field of Nonlinear Optimisation. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity. Mathematics Subject Classification: 65K99; 90C2

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Outer-space branch-and-bound algorithm for generalized linear multiplicative programs

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Optimisation stochastique des systèmes multi-réservoirs par l'agrégation de scénarios et la programmation dynamique approximative

Les problèmes de gestion des réservoirs sont stochastiques principalement à cause de l’incertitude sur les apports naturels. Ceci entraine des modèles d’optimisation de grande taille pouvant être difficilement traitables numériquement. La première partie de cette thèse réexamine la méthode d’agrégation de scénarios proposée par Rockafellar et Wets (1991). L’objectif consiste à améliorer la vitesse de convergence de l’algorithme du progressive hedgging sur lequel repose la méthode. L’approche traditionnelle consiste à utiliser une valeur fixe pour ce paramètre ou à l’ajuster selon une trajectoire choisie a priori : croissante ou décroissante. Une approche dynamique est proposée pour mettre à jour le paramètre en fonction d’information sur la convergence globale fournie par les solutions à chaque itération. Il s’agit donc d’une approche a posteriori. La thèse aborde aussi la gestion des réservoirs par la programmation dynamique stochastique. Celle-ci se prête bien à ces problèmes de gestion à cause de la nature séquentielle de leurs décisions opérationnelles. Cependant, les applications sont limitées à un nombre restreint de réservoirs. La complexité du problème peut augmenter exponentiellement avec le nombre de variables d’état, particulièrement quand l’approche classique est utilisée, i.e. en discrétisant l’espace des états de « manière uniforme ». La thèse propose une approche d’approximation sur une grille irrégulière basée sur une décomposition simpliciale de l’espace des états. La fonction de valeur est évaluée aux sommets de ces simplexes et interpolée ailleurs. À l’aide de bornes sur la vraie fonction, la grille est raffinée tout en contrôlant l’erreur d’approximation commise. En outre, dans un contexte décision-information spécifique, une hypothèse « uni-bassin », souvent utilisée par les hydrologues, est exploitée pour développer des formes analytiques pour l’espérance de la fonction de valeur. Bien que la méthode proposée ne résolve pas le problème de complexité non polynomiale de la programmation dynamique, les résultats d’une étude de cas industrielle montrent qu’il n’est pas forcément nécessaire d’utiliser une grille très dense pour approximer la fonction de valeur avec une précision acceptable. Une bonne approximation pourrait être obtenue en évaluant cette fonction uniquement en quelques points de grille choisis adéquatement.Reservoir operation problems are in essence stochastic because of the uncertain nature of natural inflows. This leads to very large optimization models that may be difficult to handle numerically. The first part of this thesis revisits the scenario aggregation method proposed by Rochafellar and Wets (1991). Our objective is to improve the convergence of the progressive hedging algorithm on which the method is based. This algorithm is based on an augmented Lagrangian with a penalty parameter that plays an important role in its convergence. The classical approach consists in using a fixed value for the parameter or in adjusting it according a trajectory chosen a priori: decreasing or increasing. This thesis presents a dynamic approach to update the parameter based on information on the global convergence provided by the solutions at each iteration. Therefore, it is an a posteriori scheme. The thesis also addresses reservoir problems via stochastic dynamic programming. This scheme is widely used for such problems because of the sequential nature of the operational decisions of reservoir management. However, dynamic programing is limited to a small number of reservoirs. The complexity may increase exponentially with the dimension of the state variables, especially when the classical approach is used, i.e. by discretizing the state space into a "regular grid". This thesis proposes an approximation scheme over an irregular grid based on simplicial decomposition of the state space. The value function is evaluated over the vertices of these simplices and interpolated elsewhere. Using bounds on the true function, the grid is refined while controlling the approximation error. Furthermore, in a specific information-decision context, a "uni-bassin" assumption often used by hydrologists is exploited to develop analytical forms for the expectation of the value function. Though the proposed method does not eliminate the non-polynomial complexity of dynamic programming, the results of an industrial case study show that it is not absolutely necessary to use a very dense grid to appropriately approximate the value function. Good approximation may be obtained by evaluating this function at few appropriately selected grid points

On the exhaustivity of simplicial partitioning

During the last 40 years, simplicial partitioning has been shown to be highly useful, including in the field of nonlinear optimization, specifically global optimization. In this article, we consider results on the exhaustivity of simplicial partitioning schemes. We consider conjectures on this exhaustivity which seem at first glance to be true (two of which have been stated as true in published articles). However, we will provide counter-examples to these conjectures. We also provide a new simplicial partitioning scheme, which provides a lot of freedom, whilst guaranteeing exhaustivity